Weak solutions of a two-phase Navier–Stokes model with a general slip law

Abstract

This work is devoted to a study of a gas–liquid Navier–Stokes model with a general slip law that allows the two phases to flow with unequal fluid velocity. The slip law is general enough to describe counter-current flow, i.e., a situation where gas and liquid move in opposite direction. Motivated by applications in the context of wellbore flow systems we assume that there is a free interface which separates the gas–liquid mixture region from a strongly gas dominated region which holds a specified positive pressure p⁎. The slip law is incorporated in the mass and momentum equations resulting in a model where flow is described in terms of the gas velocity. However, the mixture momentum equation contains a generalized pressure law composed of the standard pressure function combined with three new terms that depend on parameters characterizing the difference between gas and liquid velocity. These new terms make the analysis challenging. By carefully exploiting the positive external pressure p⁎ we are able to obtain uniform lower and upper bounds on a pressure-related quantity. This in turn allows for various higher order regularity estimates which imply that global existence and uniqueness of weak solutions can be shown.